X2055–X2119

Orion Transforms, Circumcircle Inverses, PK/NK Transforms, Reflections & Kepler Points

Orion transforms OT(X) (X2055–X2065), Caroli and Cursa points, circumcircle inverses (X2070–X2080), PK and NK transforms (X2081–X2092), classical reflections (X2093–X2105), and Kepler points and their isotomic conjugates (X2106–X2119).
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) ( A R_b ): Euclid I.22 Guide Circle (A) ( B R_a ): Euclid I.22 Guide Circle (B) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) circumcircle: Circumcircle of △ABC incircle: Incircle of △ABC A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) R_b = Radius Point R_b (0.8571, 0.0000) R_a = Radius Point R_a (0.2857, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_1 = INCENTER (0.5714, 0.2333) X_2055 = ORION TRANSFORM OF X(3) (0.3946, -0.0721) X_2056 = ORION TRANSFORM OF X(6) (0.6506, 0.2823) X_2057 = ORION TRANSFORM OF X(8) (0.2802, -0.0521) X_2058 = ORION TRANSFORM OF X(15) (0.4673, 0.0312) X_2059 = ORION TRANSFORM OF X(16) (1.0560, 0.8630) X_2060 = ORION TRANSFORM OF X(20) (0.5150, 0.1413) X_2065 = ORION TRANSFORM OF X(98) (1.2100, -0.0942) X_2066 = POINT CAROLI I (0.5679, 0.2199) X_2083 = NK-TRANSFORM OF X(19) (0.6347, 0.1449) X_2086 = PK-TRANSFORM OF X(99) (1.2449, 0.9603) X_2088 = PK-TRANSFORM OF X(110) (0.2299, -0.3440) X_2106 = POINT KEPLER I (1.5281, 1.2647) X_2108 = POINT KEPLER I (-0.3883, 0.6500) X_2110 = POINT KEPLER III (-0.2477, 0.7305) X_2112 = POINT KEPLER IV (-0.0981, 0.7800) X_2114 = POINT KEPLER V (0.1793, 0.8022) A B C X3 X1 X2055 X2056 X2057 X2058 X2059 X2060 X2065 X2066 X2083 X2086 X2088 X2106 X2108 X2110 X2112 X2114 (a, b, c) = (5, 6, 7) | AB = 1

X2055–X2119 (Orion Transforms, Circumcircle Inverses, PK/NK Transforms, Reflections & Kepler Points) — 16 representative centers (from 65 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — Orion Transforms, 2055 -2065

On September 24, 2003, Jean-Pierre Ehrmann suggested in Hyacinthos 7999 a transformation: suppose P =\(x : y : z\) (trilinears); the Orion transform of P is given by OT(P) =\(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where

\(f(a,b,c) = x[- y^{2}z^{2} + x^{2}(y^{2} + z^{2} + 2yz \cos A)]\).

Among pairs (i,j) for which OT(Xj) = Xi are these:

(1,35), (2,69), (4,24), (7,57), (99,249), (100,59), (110,250)

Ehrmann’s construction of OT(P) is as the perspector of triangles ABC and A″B″C″, where A″ is the reflection of P in the sideline B'C' of the cevian triangle A'B'C' of P, and B″ and C' are defined cyclically. Alternative and related constructions are given in other Hyacinthos messages prompted by #7999.

Group Preamble — Inverses in Circumcircle, 2070 -2080

Suppose X is a point in the plane of triangle ABC, other than the circumcenter O and not on the line at infinity. The inverse-in-circumcircle of X is the point X' on ray OX whose distance from O is \(R^{2}\)/|OX|2. In normalized trilinears, if

X = (x, y, z) O = (\(x_{3}\), \(y_{3}\), \(z_{3})\), R = circumradius,

then X' is given as (f(a,b,c), f(b,c,a), f(c,a,b)) by \(f(a,b,c) = x_{3} + (x - x_{3})(R/\)|OX|)2.

In this section, representations using the circumradius R, inradius r, Brocard angle ω and J as defined at X1113, were contributed by Peter J. C . Moses.

Group Preamble — PK and NK Transforms, 2081 - 2088

On October 1, 2003, Bernard Gibert described two transforms, given for X =\(u : v : w\) (barycentrics) by

PK(X) = \(a^{2}(c^{2}v^{2} - b^{2}w^{2})\)u : \(b^{2}(a^{2}w^{2} - c^{2}u^{2})\)v : \(c^{2}(b^{2}u^{2} - a^{2}v^{2})\)w;

NK(X) = \(a^{2}(c^{2}v^{2} + b^{2}w^{2})\)u : \(b^{2}(a^{2}w^{2} + c^{2}u^{2})\)v : \(c^{2}(b^{2}u^{2} + a^{2}v^{2})\)w.

Let \(X ^{-1}\) denote the isogonal conjugate of X. Then PK(X) is the point of intersection of the trilinear polar of X and the trilinear polar of \(X ^{-1}\), and NK(P) is the pole of the line \(XX ^{-1}\) with respect to the conic that passes through points A, B, C, X, and \(X ^{-1}\).

Also, PK(X) is the crossdifference, and NK(X) the crosssum, of X and \(X ^{-1}\).

The transform PK carries any point on the cubic pK(X6,X) to the trilinear polar of \(X ^{-1}\), and NK carries any point on the cubic nK0(X6,X) to the trilinear polar of \(X ^{-1}\). (The notations for these cubics are given at Bernard Gibert’s Cubics in the Triangle Plane.)

PK(Xi) = Xj for these (i,j):
(2,512), (3,647), (4,647), (6,512), (9,663), (19,810), (57,663), (63,810)
NK(Xi) = Xj for these (i,j):
(1,1), (2,30), (3,185), (4,185), (6,39), (98,446), (511,446)

In trilinears: suppose X =\(x : y : z\); then

PK(X) = (\(y^{2}\) - \(z^{2})\)x : (\(z^{2}\) - \(x^{2})\)y : (\(x^{2}\) - \(y^{2})\)z;

NK(X) = (\(y^{2}\) + \(z^{2})\)x : (\(z^{2}\) + \(x^{2})\)y : (\(x^{2}\) + \(y^{2})\)z.

The self-isogonal cubic pK(X6,X) is denoted as Z(X) in TCCT (p. 240), and nK0(K6,X), as \(Z^{+}(L)\). Extending examples on p. 240-241,

Z(X2) passes through Xi for I = 1,2,3,4,6,9,57, 223, 282, 1073, 1249.
Z(X4) passes through Xi for I = 1,3,4,46,90,155,254,371,372,485,486,487,488.
Z(X5) passes through Xi for I = 1,3,4,5,17,18,54,61,62,195,627,628.
Z(X20) passes through Xi for I = 1,3,4,20,40,64,84,1490,1498.
Z(X30) passes through Xi for I = 1,3,4,13,14,15,16, 30, 74, 399, 484, 616,617,1138,1157,1263,1276,1277,1337,1338.
Z(X98) passes through Xi for I = 1,98,511,1687,1688,1756,2009,2010.
Z(X99) passes through Xi for I = 1,39,83,99,512,1018,1019,1379,1380.
Z(X100) passes through Xi for I = 1,100,513,1381,1382.
Z(X110) passes through Xi for I = 1,5,54,110,523,1113,1114.
Z+(X1X6) passes through Xi for I = 1,44,88,239,241,292,294,1931.
Z+(X3X6) passes through Xi for I = 2,6,13,14,15,16,111,524
Z+(X1X2) passes through Xi for I = 1,238,291,899
Z+(X1X3) passes through Xi for I = 1,105,243,296,518,1155,1156

Additional pass-through points for these and other cubics are given at Bernard Gibert’s Cubics in the Triangle Plane.

Group Preamble — Reflections, 2093 - 2105

Suppose X =\(x : y : z\) and U =\(u : v : w\) (trilinears). Then the reflection of X in U is the point

(au - bv - cw)x + 2u(by + cz) : (bv - cw - au)y + 2v(cz + ax) : (cw - au - bv)z + 2w(ax + by).

Thus, if x = x(a,b,c), y = x(b,c,a), z = x(c,a,b) and u = u(a,b,c), v = u(b,c,a), w = u(c,a,b), then the reflection of X in U is given by \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = (au - bv - cw)x + 2u(by + cz)\).

In this section, several coordinate representations were contributed by Peter J. C . Moses.

Group Preamble — Centers 2106 -2119 (points on the 2nd equal-areas cubic, EAC2)

are on the 2nd equal-areas cubic, EAC2. (For the definition, click Tables and then Bicentric Pairs.)

For any point P on EAC2, the X2-isoconjugate of P is also on EAC2. In the following table, for each (i,j), the centers Xi and Xj are on EAC2 and are an X2-isoconjugate pair. Each pair are collinear with the pivot X238.

I J 1 6 2 31 105 672 238 292 365 365 1423 2053 1931 2054 2106 2107 2108 2109 2110 2111 2112 2113 2114 2115 2116 2117 2118 2119 2144 2145 2146 2147 3009 20332

Triangle centers on a cubic yield non-central points on the cubic; e.g., if P and P' are on EAC2, then the line of PP' meets EAC2 in third point, L(P,P'), on EAC2. (Possibly L(P,P') is P or P'.) If A', B', C' are central triangle vertices and X is a triangle center, and all four points A', B', C', and X lie on EAC2, then the points

L(A',X), L(B',X), L(C',X)

are central triangle vertices on EAC2. (A definition of central triangle is given in the Glossary.) In the following table, column 1 indicates the A-vertex of such a triangle; column 2 two points collinear with that vertex; and column 3, the A-vertex of the X2-isoconjugate of the triangle. Taking row 1 as an example, - a : c : represents the point \(-a : c : b\), which is the A-vertex of the central triangle with B-vertex \(c : - b : a\) and C-vertex \(b : a : - c\). The A-vertex is collinear with A and X2 (so that the B-vertex is collinear with B and X2, etc.). Then, in column 3, - bc : \(a^{2}\)b : represents the A-vertex of the X2-isoconjugate of - a : c : . The vertices - a : c : and - bc : \(a^{2}\)b : are collinear with the pivot, X238.

A-vertex Line A-vertex (iso-) - a : c : A, X2 - bc : \(b^{2}\) : - \(a^{1/2}\) : \(b^{1/2}\) : A, X365 - \(a^{1/2}\) : \(b^{1/2}\) : - a(a + b + c) : bc + ca + ab : A, X1 - 1/(a + b + c) : b/(bc + ca + ab) : bc + ab + ac : - (a + b + c)b : A, X6 a/(bc + ab + ac) : - 1/(a + b + c) : - bc : \(b^{2}\) : A, X31 - a : c : a(a + b + c) : \(a^{2}\) + \(b^{2}\) - (a + b)c: X6, - a : c : 1/(a + b + c) : b/[\(a^{2}\) + \(b^{2}\) - (a + b)c] : 2\(a(b^{2}\) + ca)(\(c^{2}\) + ab) :
(\(b^{2}\) + ca)(\(a^{3}\) + \(b^{3}\) - \(c^{3}\) - abc) : X31, - a : c : 1/[2\(a(b^{2}\) + ca)(\(c^{2}\) + ab)] :
b/[(\(b^{2}\) + ca)(\(a^{3}\) + \(b^{3}\) - \(c^{3}\) - abc)] : (a+b)(a+c)(\(a^{2}\)+\(b^{2}\)+\(c^{2}\)+bc+ca+ab) :
b(a+c)(\(a^{2}\)+\(b^{2}\)+ab-c(a+b+c)) : X1, - a : c : a/[(a+b)(a+c)(\(a^{2}\)+\(b^{2}\)+\(c^{2}\)+bc+ca+ab)] :
1/[(a+c)(\(a^{2}\)+\(b^{2}\)+ab-c(a+b+c))] : a(bc + ab + ac) : (c - b)\(a^{2}\) - \(b^{2}\)a + \(b^{2}\)c : X1, - bc : \(b^{2}\) : 1/[(bc + ab + ac) : b/[(c - b)\(a^{2}\) - \(b^{2}\)a + \(b^{2}\)c] : \(2a^{2}\)\(bc(b^{2}\) + ca)(\(c^{2}\) + ba) :
(\(b^{2}\) + ca)(\(b^{3}\)\(c^{3}\) + \(a^{3}\)\(c^{3}\) - \(a^{3}\)\(b^{3}\) - \(a^{2}\)\(b^{2}\)\(c^{2})\) : X2, - bc : \(b^{2}\) : 1/[2\(abc(b^{2}\) + ca)(\(c^{2}\) + ba)] :
b/[(\(b^{2}\) + ca)(\(b^{3}\)\(c^{3}\) + \(a^{3}\)\(c^{3}\) - \(a^{3}\)\(b^{3}\) - \(a^{2}\)\(b^{2}\)\(c^{2})\)] : (a+b)(a+c)[(\(b^{2}\)+bc+\(c^{2})\)\(a^{2}\) + abc(b+c) + \(b^{2}\)\(c^{2}\)] :
b(a+c)[(\(c^{2}\)-bc-\(b^{2})\)\(a^{2}\) + abc(c-b) + \(b^{2}\)\(c^{2}\)] : X6, - bc : \(b^{2}\) : 1/[(a+b)(a+c)[(\(b^{2}\)+bc+\(c^{2})\)\(a^{2}\) + abc(b+c) + \(b^{2}\)\(c^{2}\)]] :
1/[(a+c)[(\(c^{2}\)-bc-\(b^{2})\)\(a^{2}\) + abc(c-b) + \(b^{2}\)\(c^{2}\)]] :
X_2055 — ORION TRANSFORM OF X_3

X2055ORION TRANSFORM OF X3

X2055 is the perspector of triangle ABC and the orthic triangle of the cevian triangle of the circumcenter. (Randy Hutson, 9/23/2011)
X_2067 — POINT CAROLI II

X2067POINT CAROLI II

X2067 = homothetic center of 1st Kenmotu diagonals triangle and anti-tangential midarc triangle
X_2076 — INVERSE-IN-CIRCUMCIRCLE OF X_39

X2076INVERSE-IN-CIRCUMCIRCLE OF X39

Let T1 be the circumcevian triangle of the 1st Brocard point, and let T2 be the circumcevian triangle of the 2nd Brocard point. Let VAVBVC be the vertex triangle of T1 and T2, and let \(S_{A}\)\(S_{B}\)\(S_{C}\) be the side triangle of T1 and T2. The lines VA\(S_{A}\), VB\(S_{B}\), VC\(S_{C}\) concur in X2076.
X_2090 — 1st STEVANOVIC PERSPECTOR

X20901st STEVANOVIC PERSPECTOR

The bisector AX1 of angle A meets the A-excircle in two points; let A' be the point closer to I, and define B' and C' cyclically. Then triangle A'B'C' is perspective to the medial triangle, and the perspector is X2090. (Milorad Stevanovic, Hyacinthos #8088, 10/02/03)
X_2091 — 2nd STEVANOVIC PERSPECTOR

X20912nd STEVANOVIC PERSPECTOR

The bisector AX1 of angle A meets the A-excircle in two points; let A' be the point closer to I, and define B' and C' cyclically. Then triangle A'B'C' is perspective to the intouch triangle, and the perspector is X2091. (Milorad Stevanovic, Hyacinthos #8088, 10/02/03)
X_2092 — DANNEELS-APOLLONIUS PERSPECTOR

X2092DANNEELS-APOLLONIUS PERSPECTOR

Let A' be the point common to Apollonius circle and the A-excircle, and define B' and C' cyclically. Then triangle A'B'C' is perspective to the cevian triangle of X6, and the perspector is X2092. (Eric Danneels, Hyacinthos, #8070, 10/01/03). (Triangle A'B'C' is also perspective to the cevian triangle of X1; see X1682.)
X_2108 — POINT KEPLER I

X2108POINT KEPLER I

X2108 = perspector of 2nd Sharygin triangle and unary cofactor triangle of 1st Sharygin triangle